432 lines
18 KiB
Markdown
432 lines
18 KiB
Markdown
---
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Author: Antoine Foucault-Castelli
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Date: 23/03/2026
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Update: 31/03/2026
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Bibliography: @karmiFiabiliteOptimisationSystemes @ghorbelEffectBrakeLocation2020a @zhengAnalyticalApproachMesh2022
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Version: 5.0
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---
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> [!BUG]
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> Some equations has been put in latex block code as they didnt show properly on gitlab
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# Gearbox simulation
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## Structure
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```
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Functions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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-- main
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main.m
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>> used to launch all run_*.m at once
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main_parameters.m
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>> charge model's parameters
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-- runners
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run_main.m
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>> PURPOSE: launch every class_func functions in the right order to model the system
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run_display.m
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>> PURPOSE: launch every class_displayer functions
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run_print.m
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>> PURPOSE: print the figures made by run_display
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>> OUTPUT: .svg
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-- classes
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class_displayer.m
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>> displays the inputs and outputs of the model
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disp_xy(t,x,i)
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speed_xy(t,x,i)
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acc_xy(t,x,i)
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disp_theta(t,x,i)
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speed_theta(t,x,i)
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acc_theta(t,x,i)
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mesh_stiffness(t,k_var,k_mean,i)
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taero(t,T_aero,i)
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rms_on_x(x)
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spectral_on_x(x,i,tf,dt,f_mesh,N1,f_wind,ratio,maxf,labely)
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multi_spectral_on_x(x,tf,dt,maxf)
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plot_x(t,x,i)
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plot_fep(t,Fep,i)
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class_func.m
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>> model functions (explicit cf. comments in code)
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[t, dt, tf, t_mesh, f_mesh, fs] =
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time_sampling(fs_div,n_periods,Z1,N1)
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[Jin, J1, J2, Jout, J_all] =
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inertia(m_all,r_blade,di1,di2)
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[c12, db1, db2] =
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gearbox_geometry(alpha0, Z1, Z2, m0, a)
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[k_var, k_mean] =
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mesh_stiffness(epsilon, t, t_mesh, k_mesh_mean)
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[T_aero,T_out] =
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wind_excitation(t,rho_air,r_blade,v_wind,T_fluct_in,f_ext,omega_in,cp,ratio)
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[M] =
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mass_matrix(m_all, J_all)
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[K_var, K_const, K_mean] =
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stiffness_matrix(db1, db2, alpha0, kx, ky, ktheta, k_mean)
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[K] =
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stiffness_update(K_const,K_var,k)
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[C] =
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damping_matrix(M,K_mean,damp_eta,damp_beta)
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[F] =
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forces_matrix(T_in,T_out)
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[x,v,a] =
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initialization(N1,t,ratio,cp)
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[x,v,a] =
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newmark(M,C,K_const,K_var,K_mean,k_var,dt,t,T_in,T_out,x,v,a,omega_in,gamma,beta,F_ep)
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[F_ep] =
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error_profile_excitation(e12,f_mesh,t,alpha0,db1,db2,k_var)
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[damp_eta, damp_beta] =
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damping_coeff(M,K,xi1,xiN)
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-- standalone
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_convergence.m
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>> purpose: test the numerical convergence of the model
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>> output: 3 graphs + print/_convergence/*.svg
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_fft_analysis.m
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>> purpose: fft analysis for every parameter
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>> output: 6 graphs + print/_fft_analysis/*.svg
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_gear-mesh-var.m
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>> purpose: shows the variation of gear mesh with central distance changes
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>> output: 1 graph + print/_gear-mesh-var/*.svg
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_rms-analysis.m
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>> purpose: analysis of RMS, STD and kurtosis for many setups
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>> output: data/rms.svg
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```
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## How does it work ?
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**For Matlab under windows :**
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classic matlab
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> [!WARNING]
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> NOT TESTED UNDER MATLAB
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> remove all occurences of `graphics_toolkit("gnuplot")` in the code which is only useful for Octave visualization
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**For Octave/Linux :**
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*May need a few package installation (follow octave recommandations after first(s) run*
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```baswind_turbine_gearbox/octave/h
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cd wind_turbine_gearbox/octave/
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octave
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>>> run main.m
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>>>
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```
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## Informations
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- [x] Verified code with base code and paper
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- [x] Implemented wind excitation
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- [x] Implemented Ghorbel parameters
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- [~] Add varying output torque depending on number of tooth in contact ?
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- [~] add brake ?
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- [~] Variation of C ?
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- [~] ODE45 ?
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- [x] Displays from last version (in class objects)
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- [x] New mesh stiffness variation model (trapezoidal)
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- [x] Print in pdf
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- [x] Errors/defects implementation
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- [x] profile error
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- [x] assembly defect (change on gear stiffness)
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- [x] README
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- [x] Maths
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- [~] Images
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- [x] Source(s)
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- [x] Display
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- [x] time response on a2
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- [x] speed response
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- [x] x movement (vs y movement ?)
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- [x] RMS STD Kurtosis
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- [x] Printing pdf and svg
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- [x] Convergence verification
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- [x] Display effect of defects
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- [ ] Verify convergence for other input
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- [ ] Preparation Monte Carlo Parallel calculation
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- [x] Decide a number of samples per mesh period
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- [ ] Random models implementation
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## Maths
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The model is represented by two 4DDL shafts in a gear connection : $x$ and $y$ are free at the pinion, $\theta$ is free at both ends of the shaft (input/output is $\theta_{ii}$ and meshing rotation is $\theta_{ij}$).
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### Parameters
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| Newmark parameters | Value | **Unit** |
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| ------------------------------------------------ | --------- | --------------- |
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| $\gamma$ | 0.7 | |
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| $\beta$ | 0.3 | |
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| **Gears** | **Value** | |
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| Pinon 1 tooth number $Z1$ | 72 | |
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| Pinion 2 tooth number $Z2$ | 18 | |
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| Pinion modulus $m_0$ | 0.016 | m |
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| Pinion width $b$ | 0.1 | m |
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| Internal diameter pinion 1 $d_{i1}$ | 0.5 | m |
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| Internal diameter pinion 2 $d_{i_2}$ | 0.3 | m |
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| Input speed $N_1$ | 17 | m/s |
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| Contact angle $\alpha$ | 20 | degree |
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| Steel density $\rho_{steel}$ | 7860 | kg/m3 |
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| **Masses** | **Value** | **Unit** |
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| Input mass $m_{in}$ | 54000 | kg |
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| Output mass $m_{out}$ | 10 | kg |
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| **Stiffness** | | |
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| Flexion x : $k_x=k_{x_1}=k_{x_2}$ | 1e8 | N/m |
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| Flexion y : $k_y=k_{y_1}=k_{y_2}$ | 1e8 | N/m |
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| Torsion z : $k_\theta=k_{\theta_1}=k_{\theta_2}$ | 1e5 | N/m |
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| Mean mesh stiffness $k_{mesh_{mean}}$ | 1e8 | N/rad |
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| **Damping** | **Value** | **Unit** |
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| Mass damping ratio | 0.05 | |
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| Stiffness damping ratio | 0.01 | |
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| **Wind excitation** | **Value** | **Unit** |
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| Blade radius $r_{blade}$ | 6 | m |
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| Air density $\rho_{air}$ | 1.225 | kg/m3 |
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| Wind speed (mean) $v_{wind_{mean}}$ | 37.5 | m/s |
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| Fluctuating external torque $T_{fluct}$ | 50 | N |
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| Fluctuating external frequency $f_{fluct}$ | 6 | Hz |
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| Performance efficiency $cp$ | 16/27 | |
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| **Errors** | **Value** | **Unit** |
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| Profile error (approx) | 1e-5 | m |
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| Center distance error (approx) | 1e-2 | m |
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### Energy equations
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$$
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\frac{\partial Ep}{\partial q_i}+
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\frac{d}{dt}\left(\frac{Ec}{\dot{q}_i}\right)-
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\frac{\partial Ec}{\partial q_i}+
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\frac{\partial D}{\partial \dot{q}_i}
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=\frac{W}{\partial q_i}
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\\
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Ec = \frac{1}{2}\left(m_1\dot{x}_1^2+
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m_1\dot{y}_1^2+
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I_{11}\dot{\theta}_{11}^2+
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I_{12}\dot{\theta}_{12}^2+
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m_2\dot{x_2}^2+
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m_2\dot{y_2}^2+
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I_{22}\dot{\theta}_{22}^2+
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I_{21}\dot{\theta}_{21}^2\right)
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\\
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Ep = \frac{1}{2}\left( k_{x_1}x_{1}^2+
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k_{y_1}y_{1}^2+
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k_{\theta_1}(\theta_{11}-\theta_{12})^2+
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k_{x_2}x_{2}^2+
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k_{y_2}y_{2}^2+
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k_{\theta_2}(\theta_{21}-\theta_{22})^2+
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K_e(t)\delta_1^2(t)\right)
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\\
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D=
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\frac{1}{2}\left(
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D_{x_1}\dot{x_1}^2+
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D_{y_1}\dot{x_1}^2+
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D_{\theta_{11}}\dot{\theta}_{11}^2+
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D_{\theta_{12}}\dot{\theta}_{12}^2+
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D_{x_2}\dot{x_2}^2+
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D_{y_2}\dot{y_2}^2+
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D_{\theta_{22}}\dot{\theta}_{22}^2+
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D_{\theta_{21}}\dot{\theta}_{21}^2
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\right)
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$$
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### Equations of movement
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$Ep$ and $Ec$ are separated between the two shaft (1 and 2)
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```latex
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$$
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\begin{cases}
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&\frac{\partial Ep}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec}{\partial \dot{x}_1}\right)+\frac{\partial D}{\partial \dot{x}_1}
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=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)+D_{x_1}\dot{x_1}^2=0 \\
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&\frac{\partial Ep}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec}{\partial \dot{y}_1}\right)+\frac{\partial D}{\partial \dot{y}_2}
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=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)+D_{y_1}\dot{y_1}^2=0 \\
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&\frac{\partial Ep}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec}{\partial \dot{\theta}_{11}}\right)+\frac{\partial D}{\partial \dot{\theta}_{11}}
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=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})+D_{\theta_{11}}\dot{\theta}_{11}=T_{in} \\
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&\frac{\partial Ep}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec}{\partial \dot{\theta}_{12}}\right)+\frac{\partial D}{\partial \dot{\theta}_{12}}
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=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)+D_{\theta_{12}}\dot{\theta}_{12}=0 \\
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&\frac{\partial Ep}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec}{\partial \dot{x}_2}\right)+\frac{\partial D}{\partial \dot{x}_2}
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=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)+D_{x_2}\dot{x_2}^2=0 \\
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&\frac{\partial Ep}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec}{\partial \dot{y}_2}\right)+\frac{\partial D}{\partial \dot{y}_2}
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=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)+D_{y_2}\dot{y_2}^2=0 \\
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&\frac{\partial Ep}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec}{\partial \dot{\theta}_{22}}\right)+\frac{\partial D}{\partial \dot{\theta}_{22}}
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=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})+D_{\theta_{22}}\dot{\theta}_{22}=0 \\
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&\frac{\partial Ep}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec}{\partial \dot{\theta}_{21}}\right)+\frac{\partial D}{\partial \dot{\theta}_{21}}
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=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)+D_{\theta_{21}}\dot{\theta}_{21}=0
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\end{cases}
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$$
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```
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With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.
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### Matrices
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**Equation to solve :**
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$$
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[M]\{\ddot{q}\}+D\{\dot{q}\}+([K_{var}(t)]+[K_{cst}])\{q\}=\{F_0\} \\
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\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
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$$
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**Mass matrix :**
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$$
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M =
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\begin{bmatrix}
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m_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & m_1 & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & 0 & I_{11} & 0 & 0 & 0 & 0 & 0 \\
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0 & 0 & 0 & I_{12} & 0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & m_2 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & 0 & m_2 & 0 & 0 \\
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0 & 0 & 0 & 0 & 0 & 0 & I_{22} & 0 \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & I_{21} \\
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\end{bmatrix} \\
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$$
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With $m_1$ mass of shaft 1 and $m_2$ mass of shaft 2
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**Variable mesh stiffness matrix**
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$$
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K_{var}(t) = K_e(t)
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\begin{bmatrix}
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s_3 & s_5 & 0 & s_7 & -s_3 & -s_5 & 0 & s_9 \\
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s_5 & s_4 & 0 & s_6 & -s_5 & -s_4 & 0 & s_8 \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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s_7 & s_6 & 0 & s_{10} & -s_7 & -s_6 & 0 & s_{12} \\
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-s_3 & -s_5 & 0 & -s_7 & s_3 & s_5 & 0 & -s_9 \\
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-s_5 & -s_4 & 0 & -s_6 & s_5 & s_4 & 0 & -s_8 \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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s_9 & s_8 & 0 & s_{12} & -s_9 & -s_8 & 0 & s_{11}
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\end{bmatrix}\\
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\text{with} \\
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s_1 =\sin(\alpha) \hspace{1cm}
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s_2 =\cos(\alpha) \hspace{1cm}
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s_3 =\sin^2(\alpha) \\
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s_4 =\cos^2(\alpha) \hspace{1cm}
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s_5 =\sin(\alpha)\cos(\alpha) \hspace{1cm}
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s_6 =r_1\cos(\alpha) \\
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s_7 =r_1\sin(\alpha) \hspace{1cm}
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s_8 =r_2\cos(\alpha) \hspace{1cm}
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s_9 =r_2\sin(\alpha) \\
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s_{10} =r_1^2 \hspace{1cm}
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s_{11} =r_2^2 \hspace{1cm}
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s_{12} =r_1r_2
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$$
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> [!NOTE]
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>
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> In matlab, cos and sin terms will be noted as $s_i$, for visual simplification
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**Constant stiffness matrix**
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$$
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Ks=
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\begin{bmatrix}
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k_{x1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & k_{y1} & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & 0 & k_{\theta1}& -k_{\theta1} & 0 & 0 & 0 & 0 \\
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0 & 0 & -k_{\theta1} & k_{\theta1} & 0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & k_{x1} & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & 0 & k_{y1} & 0 & 0 \\
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0 & 0 & 0 & 0 & 0 & 0 & k_{\theta2} & -k_{\theta2} \\
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0 & 0 & 0 & 0 & 0 & 0 & -k_{\theta2} & k_{\theta2} \\
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\end{bmatrix}
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$$
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**Damping matrix**
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Damping matrix is : $C = \eta M+\beta K_{mean}$
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**Forces**
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$$
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F =
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\begin{bmatrix}
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0 \\ 0 \\ T_{aero} \\ 0 \\ 0\\ 0\\ 0\\ 0
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\end{bmatrix}
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$$
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**Mesh stiffness model**
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The mesh stiffness is modelized by a trapezoïdal signal between a low value ($=k_{mesh_{mean}}$ for one tooth in contact) and its double value as a high value. Length of both phase is given by $T_{high}=(\varepsilon-1)T_{mesh}$ and $T_{low}=(2-\varepsilon)T_{mesh}$. 10% of the start and of the end of the “high” phase is a linear slope from low value to high value.
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> [!NOTE]
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> $T_{mesh}=1/f_{mesh}=1/(Z1*N1/60)$
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> $\varepsilon$ is an approximation using the following formula :
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```latex
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$$
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c_1=\frac{1+\frac{1}{Z1}}{\left(\pi cos(\alpha)\left(\frac{sin(\alpha)}{2}+\sqrt{\frac{sin(\alpha)^2}{4}+\frac{1}{Z1^2}+\frac{1}{Z1}}\right)\right)} \\
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c_1=\frac{1+\frac{1}{Z2}}{\left(\pi cos(\alpha)\left(\frac{sin(\alpha)}{2}+\sqrt{\frac{sin(\alpha)^2}{4}+\frac{1}{Z2^2}+\frac{1}{Z2}}\right)\right)} \\
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c_{12}=c_1+c_2 \approx \varepsilon
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$$
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```
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## Model
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**Time :** time is sampled on the $T_{mesh}$, with $fs$ the number of points for each period, and $n_{periods}$ the number of periods calculated. The number of points is $fs\times n_{periods}$.
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**Speed :** rotational speed is initialized at its “stable” value.
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> if no automatic implementation, it has to be done “by hand” by executing the program once, get the speeds after the transient zone, and put them by hand in “initialization” source code (`func.m`)
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**Wind :** cosine variation around mean torque value
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$$
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T_{in} = \rho_{air}\pi(r_{blade}^2)(v_{wind}^3)*Cp)/(2\omega) \\
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T_{aero}(t) = T_{in} + T_{fluct}\cos(2\pi f_{ext}t)
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$$
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**Initialization :** everything is set at 0 for initialization (except speed)
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**Model :** a basic Newmark scheme is used to compute the model over time
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**Error implementation :**
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- error profile is implemented as ($+\infty$ is approximated as $1000$) :
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$$
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ep(t) = e_{12}+\sum ^{+\infty}_{n=1}e_{12}sin(2n\pi f_m t) \\
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\left\{F_{ep}\right\} = \frac{\partial \delta (t)}{\partial q_i}k(t)ep(t) \\
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\frac{\partial \delta(t)}{\delta q_i} =
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\begin{bmatrix}
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sin(\alpha) \\ cos(\alpha) \\ 0 \\ rb_1 \\ -sin(\alpha) \\ -cos(\alpha) \\ 0 \\ rb_2
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\end{bmatrix}
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$$
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- center distance error is implemented as a transformation of $\alpha$ depending on a variation of $a$, in `gearbox_geometry`. It does only change the values of contact ratio $\varepsilon$ and then the profile of the mesh stiffness (length of phases and max/min values).
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$$
|
|
\alpha'=acos\left(\frac{rb_1+rb_2}{E+a}\right)
|
|
$$
|
|
|
|
|
|
|
|
|
|
## TO FIX
|
|
|
|
> [!CAUTION]
|
|
> ...
|
|
|
|
## Machine specifications
|
|
Example of computational time : `44sec for fs=2500` / `n_periods = 10`
|
|
```
|
|
-`
|
|
.o+`
|
|
`ooo/ OS: Arch Linux x86_64
|
|
`+oooo: Host: 20EV000UFR ThinkPad E560
|
|
`+oooooo: Kernel: 6.19.8-arch1-1
|
|
-+oooooo+: Shell: zsh 5.9
|
|
`/:-:++oooo+: Terminal: kitty
|
|
`/++++/+++++++: CPU: Intel i5-6200U (4) @ 2.300GHz
|
|
`/++++++++++++++: GPU: Intel Skylake-U GT2 [HD Graphics 520]
|
|
`/+++ooooooooooooo/` Memory: 1703MiB / 3792MiB
|
|
./ooosssso++osssssso+`
|
|
.oossssso-````/ossssss+`
|
|
-osssssso. :ssssssso.
|
|
:osssssss/ osssso+++.
|
|
/ossssssss/ +ssssooo/-
|
|
`/ossssso+/:- -:/+osssso+-
|
|
`+sso+:-` `.-/+oso:
|
|
`++:. `-/+/
|
|
.` `/
|
|
```
|